In some respects, the biology of the fungi family is more closely related to the animal kingdom than plants; in reality though it's more of a hybrid or even a separate division in its own right - another point in the triangle of terrestrial biology.
There is not a huge set of fungal species that in fact lives in a multi-cellular construct and those that do, procreate via spores. This is an important point because this evolutionary design makes it far less likely that the larger species of fungus need to be ambulatory. They merely spread through spores, which has been a very successful strategy for them but means that their success is heavily reliant on the next generation at every turn.
So, for your mushroom men to evolve, firstly you need an environment that encourages the fungus into a longer life cycle, but still has challenging environmental changes. Animals can take advantage of larger territories, exploiting the change in one region (say an apple tree ripening its fruit) at one point and then when that is exhausted, moving on to another (say salmon swimming upstream in a near by creek). They have far less children when compared to a fungus' spore release, but have longer lives of higher utility and variability. Evolutionarily speaking, you need the same drivers to be present with fungi to make your mushroom men happen.
Of course, you have another issue as well; senses. Animals formed a sense of smell, and later taste, touch, vision & hearing so that they could negotiate and navigate their environment when they became ambulatory. As a consequence, they could seek out food, opportunity, and sense danger in time to avoid it. Brains are useful for processing all this sensory input, but not necessary. All you need is a central nervous system of some kind for rudimentary instinctive responses to the environment to make all that work. The trouble is, even a central nervous system is a complex design to encode in DNA. Mushrooms at their core are simple constructs and evolution prefers simplicity where it can get away with it so sensory organs evolving in mushrooms are only going to happen in response to becoming ambulatory.
Brains are a whole other matter and the likelihood of two competing animal like constructs, one the animal kingdom we currently know and one based on fungal structures growing and evolving in parallel, competing for the same resources over hundreds of millions of years, is unlikely. One would get wiped out by the other at some point during which resources are scarce and the fittest evolutionary model succeeds the other.
But; it's possible. Imagine (if you will) that fungi come up from the waters first, alongside plants, and that animal progression to land is delayed even a little. If the fungi evolved into an ambulatory species on land directly, their senses would be evolved perfectly for the land. They would have a distinct advantage over traditional animals, particularly in sight, in the first instance because fish and amphibian eyes are designed for a water medium; it's going to take time for animal eyes to adapt to use on land and in an atmospheric environment. But, animals would eventually have an advantage through the fact that they are used to predators and a lack of safety (not really any places to hide in the open ocean) so they are going to be more aggressive and better at evading predators in the main. In such an environment, it's possible that when the land on earth is split into two main continents, the fungal life succeeds on one, and animal life on the other. As they split and merge, certain species of both die out, but for the most part species of each also survive.
I must stress this is all pure speculation, but it's possible. What is needed? Well, firstly a driving advantage in the environment towards a longer, ambulatory life where spores (say) can't be both light AND carry all the genetic information required for replication. Secondly, time. Lots of time to hone and refine the new ambulatory fungus and its senses in a manner that allows it to function efficiently on a land environment.
Finally though, your mushroom creatures may be close to animals, but its HIGHLY unlikely that they will be men with highly intelligent brains. I doubt they would be able to talk with you. Again its possible, but brain power takes a massive amount of energy to maintain and I suspect that evolution isn't going to support two different species with equivalent brain power on one planet - humans for instance are just too good at wiping out all their threats and I suspect that your mushroom men would be the same, meaning at some point that fight would come to a head and you'd be down to one again soon enough.